Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams

Authors Chenglin Fan, Benjamin Raichel

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Chenglin Fan
  • Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA
Benjamin Raichel
  • Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA


The authors want to thank Sariel Har-Peled for useful discussions concerning the case of sampled site locations. Also, thank you to the reviewers for their helpful comments.

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Chenglin Fan and Benjamin Raichel. Linear Expected Complexity for Directional and Multiplicative Voronoi Diagrams. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


While the standard unweighted Voronoi diagram in the plane has linear worst-case complexity, many of its natural generalizations do not. This paper considers two such previously studied generalizations, namely multiplicative and semi Voronoi diagrams. These diagrams both have quadratic worst-case complexity, though here we show that their expected complexity is linear for certain natural randomized inputs. Specifically, we argue that the expected complexity is linear for: (1) semi Voronoi diagrams when the visible direction is randomly sampled, and (2) for multiplicative diagrams when either weights are sampled from a constant-sized set, or the more challenging case when weights are arbitrary but locations are sampled from a square.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Voronoi Diagrams
  • Expected Complexity
  • Computational Geometry


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